Sometimes having the old paint underneath causes the repainted figure to become sticky overtime. Most of the times, however, I, too, have found it isn't necessary to remove the old paint. If you want to do that, however, oven cleaner suitably diluted, rubbing alcohol or even plain acetone (nail polish remover) usually work. If you use any of those raw, you risk them melting not just the paint but the plastic underneath, though, so it's worth the bother to dilute them with water. Also, as ever, before priming or painting the figurine, you'll need to wash it well before starting so that there isn't any residue or grease left on it when you start.

My trouble with the nail polish remover is that the acetone breaks down the plastic and makes it soft. It can also remove details if left on too long, as it basically "eats" the plastic. So proceed with caution